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Hindawi Publishing CorporationInternational Journal of Mathematics and Mathematical SciencesVolume 2012, Article ID 730962, 13 pagesdoi:10.1155/2012/730962
Research ArticleComplete Moment Convergence of Weighted Sumsfor Arrays of Rowwise ϕ-Mixing Random Variables
Ming Le Guo
School of Mathematics and Computer Science, Anhui Normal University, Wuhu 241003, China
Correspondence should be addressed to Ming Le Guo, [email protected]
Received 5 June 2012; Accepted 20 August 2012
Academic Editor: Mowaffaq Hajja
Copyright q 2012 Ming Le Guo. This is an open access article distributed under the CreativeCommons Attribution License, which permits unrestricted use, distribution, and reproduction inany medium, provided the original work is properly cited.
The complete moment convergence of weighted sums for arrays of rowwise ϕ-mixing randomvariables is investigated. By using moment inequality and truncation method, the sufficientconditions for complete moment convergence of weighted sums for arrays of rowwise ϕ-mixingrandom variables are obtained. The results of Ahmed et al. (2002) are complemented. As anapplication, the complete moment convergence of moving average processes based on a ϕ-mixingrandom sequence is obtained, which improves the result of Kim et al. (2008).
1. Introduction
Hsu and Robbins [1] introduced the concept of complete convergence of {Xn}. A sequence{Xn, n = 1, 2, . . .} is said to converge completely to a constant C if
∞∑
n=1
P(|Xn − C| > ε) < ∞, ∀ε > 0. (1.1)
Moreover, they proved that the sequence of arithmetic means of independent identicallydistributed (i.i.d.) random variables converge completely to the expected value if the varianceof the summands is finite. The converse theorem was proved by Erdos [2]. This result hasbeen generalized and extended in several directions, see Baum and Katz [3], Chow [4], Gut[5], Taylor et al. [6], and Cai and Xu [7]. In particular, Ahmed et al. [8] obtained the followingresult in Banach space.
2 International Journal of Mathematics and Mathematical Sciences
Theorem A. Let {Xni; i ≥ 1, n ≥ 1} be an array of rowwise independent random elements in aseparable real Banach space (B, ‖ · ‖). Let P(‖Xni‖ > x) ≤ CP(|X| > x) for some random variable X,constant C and all n, i and x > 0. Suppose that {ani, i ≥ 1, n ≥ 1} is an array of constants such that
supi≥1
|ani| = O(n−r), for some r > 0,
∞∑
i=1
|ani| = O(nα), for some α ∈ [0, r).(1.2)
Let β be such that α + β /= − 1 and fix δ > 0 such that 1 + α/r < δ ≤ 2. Denote s = max(1 + (α + β +1)/r, δ). If E|X|s < ∞ and Sn =
∑∞i=1 aniXni → 0 in probability, then
∑∞n=1 n
βP(‖Sn‖ > ε) < ∞for all ε > 0.
Chow [4] established the following refinement which is a complete momentconvergence result for sums of (i.i.d.) random variables.
Theorem B. Let EX1 = 0, 1 ≤ p < 2 and r ≥ p. Suppose that E[|X1|r + |X1| log(1 + |X1|)] < ∞.Then
∞∑
n=1
n(r/p)−2−(1/p)E
(∣∣∣∣∣
n∑
i=1
Xi
∣∣∣∣∣ − εn1/p
)+
< ∞, ∀ε > 0. (1.3)
The main purpose of this paper is to discuss again the above results for arrays ofrowwise ϕ-mixing random variables. The author takes the inspiration in [8] and discussesthe complete moment convergence of weighted sums for arrays of rowwise ϕ-mixing randomvariables by applying truncation methods. The results of Ahmed et al. [8] are extended to ϕ-mixing case. As an application, the corresponding results of moving average processes basedon a ϕ-mixing random sequence are obtained, which extend and improve the result of Kimand Ko [9].
For the proof of the main results, we need to restate a few definitions and lemmas foreasy reference. Throughout this paper, Cwill represent positive constants, the value of whichmay change from one place to another. The symbol I(A) denotes the indicator function of A;[x] indicates the maximum integer not larger than x. For a finite set B, the symbol �B denotesthe number of elements in the set B.
Definition 1.1. A sequence of random variables {Xi, 1 ≤ i ≤ n} is said to be a sequence ofϕ-mixing random variables, if
ϕ(m) = supk≥1
{|P(B | A) − P(B)| ;A ∈ k
1 , B ∈ ∞k+m, P(A) > 0
}−→ 0, as m −→ ∞, (1.4)
where kj = σ{Xi; j ≤ i ≤ k}, 1 ≤ j ≤ k ≤ ∞.
Definition 1.2. A sequence {Xn, n ≥ 1} of random variables is said to be stochasticallydominated by a random variable X (write {Xi} ≺ X) if there exists a constant C, such thatP{|Xn| > x} ≤ CP{|X| > x} for all x ≥ 0 and n ≥ 1.
International Journal of Mathematics and Mathematical Sciences 3
The following lemma is a well-known result.
Lemma 1.3. Let the sequence {Xn, n ≥ 1} of random variables be stochastically dominated by arandom variable X. Then for any p > 0, x > 0
E|Xn|pI(|Xn| ≤ x) ≤ C[E|X|pI(|X| ≤ x) + xpP{|X| > x}], (1.5)
E|Xn|pI(|Xn| > x) ≤ CE|X|pI(|X| > x). (1.6)
Definition 1.4. A real-valued function l(x), positive andmeasurable on [A,∞) for someA > 0,is said to be slowly varying if limx→∞l(xλ)/l(x) = 1 for each λ > 0.
By the properties of slowly varying function, we can easily prove the following lemma.Here we omit the details of the proof.
Lemma 1.5. Let l(x) > 0 be a slowly varying function as x → ∞, then there exists C (depends onlyon r) such that
(i) Ckr+1l(k) ≤∑kn=1 n
rl(n) ≤ Ckr+1l(k) for any r > −1 and positive integer k,
(ii) Ckr+1l(k) ≤∑∞n=k n
rl(n) ≤ Ckr+1l(k) for any r < −1 and positive integer k.
The following lemma will play an important role in the proof of our main results. Theproof is due to Shao [10].
Lemma 1.6. Let {Xi, 1 ≤ i ≤ n} be a sequence of ϕ-mixing random variables with mean zero. Supposethat there exists a sequence {Cn} of positive numbers such that E(
∑k+mi=k+1 Xi)
2 ≤ Cn for any k ≥ 0, n ≥1, m ≤ n. Then for any q ≥ 2, there exists C = C(q, ϕ(·)) such that
Emax1≤j≤n
∣∣∣∣∣∣
k+j∑
i=k+1
Xi
∣∣∣∣∣∣
q
≤ C
[C
q/2n + E max
k+1≤i≤k+n|Xi|q
]. (1.7)
Lemma 1.7. Let {Xi, 1 ≤ i ≤ n} be a sequence of ϕ-mixing random variables with∑∞
i=1 ϕ1/2(i) < ∞,
then there exists C such that for any k ≥ 0 and n ≥ 1
E
(k+n∑
i=k+1
Xi
)2
≤ Ck+n∑
i=k+1
EX2i . (1.8)
4 International Journal of Mathematics and Mathematical Sciences
Proof. By Lemma 5.4.4 in [11] and Holder’s inequality, we have
E
(k+n∑
i=k+1
Xi
)2
=k+n∑
i=k+1
EX2i + 2
∑
k+1≤i<j≤k+nEXiXj
≤k+n∑
i=k+1
EX2i + 4
∑
k+1≤i<j≤k+nϕ1/2(j − i
)(EX2
i
)1/2(EX2
j
)1/2
≤k+n∑
i=k+1
EX2i + 2
k+n−1∑
i=k+1
k+n∑
j=i+1
ϕ1/2(j − i)(
EX2i + EX2
j
)
≤(1 + 4
n∑
i=1
ϕ1/2(i)
)k+n∑
i=k+1
EX2i .
(1.9)
Therefore, (1.8) holds.
2. Main Results
Now we state our main results. The proofs will be given in Section 3.
Theorem 2.1. Let {Xni, i ≥ 1, n ≥ 1} be an array of rowwise ϕ-mixing random variables with EXni =0, {Xni} ≺ X and
∑∞m=1 ϕ
1/2(m) < ∞. Let l(x) > 0 be a slowing varying function, and {ani, i ≥ 1, n ≥1} be an array of constants such that
supi≥1
|ani| = O(n−r), for some r > 0,
∞∑
i=1
|ani| = O(nα), for some α ∈ [0, r).(2.1)
(a) If α + β + 1 > 0 and there exists some δ > 0 such that (α/r) + 1 < δ ≤ 2, and s =max(1 + ((α + β + 1)/r), δ), then E|X|sl(|X|1/r) < ∞ implies
∞∑
n=1
nβl(n)E
[supk≥1
∣∣∣∣∣
k∑
i=1
aniXni
∣∣∣∣∣ − ε
]+< ∞, ∀ε > 0. (2.2)
(b) If β = −1, α > 0, then E|X|1+(a/r)(1 + l(|X|1/r)) < ∞ implies
∞∑
n=1
n−1l(n)E
[supk≥1
∣∣∣∣∣
k∑
i=1
aniXni
∣∣∣∣∣ − ε
]+< ∞, ∀ε > 0. (2.3)
International Journal of Mathematics and Mathematical Sciences 5
Remark 2.2. If α + β + 1 < 0, then E|X| < ∞ implies that (2.2) holds. In fact,
∞∑
n=1
nβl(n)E
[supk≥1
∣∣∣∣∣
k∑
i=1
aniXni
∣∣∣∣∣ − ε
]+≤
∞∑
n=1
nβl(n)∞∑
i=1
|ani|E|Xni| + ε∞∑
n=1
nβl(n)
≤ C∞∑
n=1
nβ+αl(n)E|X| + ε∞∑
n=1
nβl(n) < ∞.
(2.4)
Remark 2.3. Note that
∞ >∞∑
n=1
nβl(n)E
[supk≥1
∣∣∣∣∣
k∑
i=1
aniXni
∣∣∣∣∣ − ε
]+=
∞∑
n=1
nβl(n)∫∞
0P
{supk≥1
∣∣∣∣∣
k∑
i=1
aniXni
∣∣∣∣∣ − ε > x
}dx
=∫∞
0
∞∑
n=1
nβl(n)P
{supk≥1
∣∣∣∣∣
k∑
i=1
aniXni
∣∣∣∣∣ > x + ε
}dx.
(2.5)
Therefore, from (2.5), we obtain that the complete moment convergence implies the completeconvergence, that is, under the conditions of Theorem 2.1, result (2.2) implies
∞∑
n=1
nβl(n)P
{supk≥1
∣∣∣∣∣
k∑
i=1
aniXni
∣∣∣∣∣ > ε
}< ∞, (2.6)
and (2.3) implies
∞∑
n=1
n−1l(n)P
{supk≥1
∣∣∣∣∣
k∑
i=1
aniXni
∣∣∣∣∣ > ε
}< ∞. (2.7)
Corollary 2.4. Under the conditions of Theorem 2.1,
(1) if α + β + 1 > 0 and there exists some δ > 0 such that (α/r) + 1 < δ ≤ 2, and s =max(1 + ((α + β + 1)/r), δ), then E|X|sl(|X|1/r) < ∞ implies
∞∑
n=1
nβl(n)E
[∣∣∣∣∣
∞∑
i=1
aniXni
∣∣∣∣∣ − ε
]+< ∞, ∀ε > 0, (2.8)
(2) if β = −1, α > 0, then E|X|1+(α/r)(1 + l(|X|1/r)) < ∞ implies
∞∑
n=1
n−1l(n)E
[∣∣∣∣∣
∞∑
i=1
aniXni
∣∣∣∣∣ − ε
]+< ∞, ∀ε > 0. (2.9)
Corollary 2.5. Let {Xni, i ≥ 1, n ≥ 1} be an array of rowwise ϕ-mixing random variables withEXni = 0,{Xni} ≺ X and
∑∞m=1 ϕ
1/2(m) < ∞. Suppose that l(x) > 0 is a slowly varying function.
6 International Journal of Mathematics and Mathematical Sciences
(1) Let p > 1 and 1 ≤ t < 2. If E|X|ptl(|X|t) < ∞, then
∞∑
n=1
np−2−(1/t)l(n)E
[max1≤k≤n
∣∣∣∣∣
k∑
i=1
Xni
∣∣∣∣∣ − εn1/t
]+< ∞, ∀ε > 0. (2.10)
(2) Let 1 < t < 2. If E|X|t[1 + l(|X|t)] < ∞, then
∞∑
n=1
n−1−(1/t)l(n)E
[max1≤k≤n
∣∣∣∣∣
k∑
i=1
Xni
∣∣∣∣∣ − εn1/t
]+< ∞, ∀ε > 0. (2.11)
Corollary 2.6. Suppose that Xn =∑∞
i=−∞ ai+nYi, n ≥ 1, where {ai,−∞ < i < ∞} is a sequence ofreal numbers with
∑∞−∞ |ai| < ∞, and {Yi,−∞ < i < ∞} is a sequence of ϕ-mixing random variables
with EYi = 0, {Yi} ≺ Y and∑∞
m=1 ϕ1/2(m) < ∞. Let l(x) be a slowly varying function.
(1) Let 1 ≤ t < 2, r ≥ 1 + (t/2). If E|Y |r l(|Y |t) < ∞, then
∞∑
n=1
n(r/t)−2−(1/t)l(n)E
[∣∣∣∣∣
n∑
i=1
Xi
∣∣∣∣∣ − εn1/t
]+< ∞, ∀ε > 0. (2.12)
(2) Let 1 < t < 2. If E|Y |t[1 + l(|Y |t)] < ∞, then
∞∑
n=1
n−1−(1/t)l(n)E
[∣∣∣∣∣
n∑
i=1
Xi
∣∣∣∣∣ − εn1/t
]+< ∞, ∀ε > 0. (2.13)
Remark 2.7. Corollary 2.6 obtains the result about the complete moment convergence ofmoving average processes based on a ϕ-mixing random sequencewith different distributions.We extend the results of Chen et al. [12] from the complete convergence to the completemoment convergence. The result of Kim and Ko [9] is a special case of Corollary 2.6 (1).Moreover, our result covers the case of r = t, which was not considered by Kim and Ko.
3. Proofs of the Main Results
Proof of Theorem 2.1. Without loss of generality, we can assume
supi≥1
|ani| ≤ n−r ,∞∑
i=1
|ani| ≤ nα. (3.1)
International Journal of Mathematics and Mathematical Sciences 7
Let Snk(x) =∑k
i=1 aniXniI(|aniXni| ≤ n−rx) for any k ≥ 1, n ≥ 1, and x ≥ 0. First note thatE|X|sl(|X|1/r) < ∞ implies E|X|t < ∞ for any 0 < t < s. Therefore, for x > nr ,
x−1nrsupk≥1
E|Snk(x)| = x−1nrsupk≥1
E
∣∣∣∣∣
k∑
i=1
aniXniI(|aniXni| > n−rx
)∣∣∣∣∣ (EXni = 0)
≤∞∑
i=1
E|aniXni|I(|aniXni| > n−rx
) ≤∞∑
i=1
E|aniX|I(|aniX| > n−rx)
≤∞∑
i=1
|ani|E|X|I(|X| > x) ≤ nαE|X|I(|X| > x)
≤ xα/rE|X|I(|X| > x) ≤ E|X|1+(α/r)I(|X| > nr) −→ 0 as n −→ ∞.
(3.2)
Hence, for n large enough we have supk≥1E|Snk(x)| < (ε/2)n−rx. Then
∞∑
n=1
nβl(n)E
[supk≥1
∣∣∣∣∣
k∑
i=1
aniXni
∣∣∣∣∣ − ε
]+
=∞∑
n=1
nβl(n)∫∞
ε
P
{supk≥1
∣∣∣∣∣
k∑
i=1
aniXni
∣∣∣∣∣ ≥ x
}dx
=∞∑
n=1
nβ−r l(n)ε∫∞
nr
P
{supk≥1
∣∣∣∣∣
k∑
i=1
aniXni
∣∣∣∣∣ ≥ εn−rx
}dx
≤ C∞∑
n=1
nβ−r l(n)∫∞
nr
P
{sup
i
|aniXni| > n−rx
}dx
+ C∞∑
n=1
nβ−r l(n)∫∞
nr
P
{supk≥1
|Snk(x) − ESnk(x)| ≥ n−rxε
2
}dx := I1 + I2.
(3.3)
Noting that α + β > −1, by Lemma 1.5, Markov inequality, (1.6), and (3.1), we have
I1 ≤ C∞∑
n=1
nβ−r l(n)∫∞
nr
∞∑
i=1
P{|aniXni| > n−rx
}dx
≤ C∞∑
n=1
nβ−r l(n)∫∞
nr
nrx−1∞∑
i=1
E|aniXni|I(|aniXni| > n−rx
)dx
≤ C∞∑
n=1
nβ+αl(n)∫∞
nr
x−1E|X|I(|X| > x)dx
≤ C∞∑
n=1
nβ+αl(n)∞∑
k=n
∫kr+1
kr
x−1E|X|I(|X| > x)dx
8 International Journal of Mathematics and Mathematical Sciences
≤ C∞∑
n=1
nβ+αl(n)∞∑
k=n
k−1E|X|I(|X| > kr) ≤ C∞∑
k=1
k−1E|X|I(|X| > kr)k∑
n=1
nβ+αl(n)
≤ C∞∑
k=1
kβ+αl(k)E|X|I(|X| > kr) ≤ CE|X|1+((1+α+β)/r)l(|X|1/r
)< ∞.
(3.4)
Now we estimate I2, noting that∑∞
m=1 ϕ1/2(m) < ∞, by Lemma 1.7, we have
sup1≤m<∞
E
(m∑
i=1
aniXniI(|aniXni| ≤ n−rx
) − Em∑
i=1
aniXniI(|aniXni| ≤ n−rx
))2
≤ C∞∑
i=1
Ea2niX
2niI(|aniXni| ≤ n−rx
).
(3.5)
By Lemma 1.6, Markov inequality, Cr inequality, and (1.5), for any q ≥ 2, we have
P
{supk≥1
|Snk(x) − ESnk(x)| ≥ n−rxε
2
}≤ Cnrqx−qEsup
k≥1|Snk(x) − ESnk(x)|q
≤ Cnrqx−q
⎡
⎣( ∞∑
i=1
Ea2niX
2niI(|aniXni| ≤ n−rx
))q/2
+∞∑
i=1
E|aniXni|qI(|aniXni| ≤ n−rx
)⎤
⎦
≤ Cnrqx−q( ∞∑
i=1
Ea2niX
2I(|aniX| ≤ n−rx
))q/2
+ Cnrqx−q∞∑
i=1
E|aniX|qI(|aniX| ≤ n−rx)
+ C
( ∞∑
i=1
P{|aniX| > n−rx
})q/2
+ C∞∑
i=1
P{|aniX| > n−rx
}
:= J1 + J2 + J3 + J4.
(3.6)
So,
I2 ≤∞∑
n=1
nβ−r l(n)∫∞
nr
(J1 + J2 + J3 + J4)dx. (3.7)
From (3.4), we have∑∞
n=1 nβ−r l(n)
∫∞nr J4dx < ∞.
For J1, we consider the following two cases.
International Journal of Mathematics and Mathematical Sciences 9
If s > 2, then EX2 < ∞. Taking q ≥ 2 such that β + (q(α − r)/2) < −1, we have
∞∑
n=1
nβ−r l(n)∫∞
nr
J1dx
≤ C∞∑
n=1
nβ−r+rql(n)∫∞
nr
x−q( ∞∑
i=1
a2ni
)q/2
dx
≤ C∞∑
n=1
nβ−r+rql(n)nq(α−r)/2nr(−q+1) ≤ C∞∑
n=1
nβ+(q(α−r)/2)l(n) < ∞.
(3.8)
If s ≤ 2, we choose s′ such that 1+ (α/r) < s′ < s. Taking q ≥ 2 such that β+ (qr/2)(1+ (α/r)−s′) < −1, we have
∞∑
n=1
nβ−r l(n)∫∞
nr
J1dx
≤ C∞∑
n=1
nβ−r+rql(n)∫∞
nr
x−q( ∞∑
i=1
|ani||ani|s′−1E|aniX|2−s′ |X|s′I(|aniX| ≤ n−rx
))q/2
dx
≤ C∞∑
n=1
nβ−r+rql(n)nqα/2n−(qr/2)(s′−1)∫∞
nr
x−q(n−rx)(q/2)(2−s′)
dx
≤ C∞∑
n=1
nβ+(qr/2)(1+(α/r)−s′)l(n) < ∞.
(3.9)
So,∑∞
n=1 nβ−r l(n)
∫∞nr J1dx < ∞.
Now, we estimate J2. Set Inj = {i ≥ 1 | (n(j + 1))−r < |ani| ≤ (nj)−r}, j = 1, 2, . . .. Then∪j≥1Inj = N, where N is the set of positive integers. Note also that for all k ≥ 1, n ≥ 1,
nα ≥∞∑
i=1
|ani| =∞∑
j=1
∑
i∈Inj|ani|
≥∞∑
j=1
(�Inj)(n(j + 1
))−r ≥ n−r∞∑
j=k
(�Inj)(j + 1
)−rq(k + 1)rq−r .
(3.10)
Hence, we have
∞∑
j=k
(�Inj)j−rq ≤ Cnα+rkr−rq. (3.11)
10 International Journal of Mathematics and Mathematical Sciences
Note that
∞∑
n=1
nβ−r l(n)∫∞
nr
J2dx
= C∞∑
n=1
nβ−r+rql(n)∫∞
nr
x−q∞∑
j=1
∑
i∈InjE|aniX|qI(|aniX| ≤ n−rx
)dx
= C∞∑
n=1
nβ−r+rql(n)∞∑
j=1
(�Inj)(nj)−rq ∞∑
k=n
∫ (k+1)r
kr
x−qE|X|qI(|X| ≤ x(j + 1
)r)dx
≤ C∞∑
n=1
nβ−r+rql(n)∞∑
j=1
(�Inj)(nj)−rq ∞∑
k=n
kr(−q+1)−1E|X|qI(|X| ≤ (k + 1)r(j + 1
)r)
= C∞∑
n=1
nβ−r l(n)∞∑
k=n
kr(−q+1)−1∞∑
j=1
(�Inj)j−rq
(k+1)(j+1)−1∑
i=0
E|X|qI(ir < |X| ≤ (i + 1)r)
≤ C∞∑
n=1
nβ−r l(n)∞∑
k=n
kr(−q+1)−1∞∑
j=1
(�Inj)j−rq
2(k+1)−1∑
i=0
E|X|qI(ir < |X| ≤ (i + 1)r)
+ C∞∑
n=1
nβ−r l(n)∞∑
k=n
kr(−q+1)−1∞∑
j=1
(�Inj)j−rq
(k+1)(j+1)∑
i=2(k+1)
E|X|qI(ir < |X| ≤ (i + 1)r)
:= J ′2 + J ′′2 .
(3.12)
Taking q ≥ 2 large enough such that β + α − rq + r < −1, for J ′2, by Lemma 1.6 and (3.11), weget
J ′2 ≤ C∞∑
n=1
nβ−r l(n)∞∑
k=n
kr(−q+1)−1nα+r2(k+1)−1∑
i=0
E|X|qI(ir < |X| ≤ (i + 1)r)
= C∞∑
k=1
kr(−q+1)−12(k+1)−1∑
i=0
E|X|qI(ir < |X| ≤ (i + 1)r) k∑
n=1
nβ+αl(n)
≤ C∞∑
k=1
kβ+α−rq+r l(k)2(k+1)−1∑
i=0
E|X|qI(ir < |X| ≤ (i + 1)r)
≤ C + C∞∑
i=3
E|X|qI(ir < |X| ≤ (i + 1)r) ∞∑
k=[i/2]
kβ+α−rq+r l(k)
≤ C + C∞∑
i=3
iβ+α−rq+r+1l(i)E|X|qI(ir < |X| ≤ (i + 1)r) ≤ C + CE|X|1+((β+α+1)/r)l
(|X|1/r
)< ∞.
(3.13)
International Journal of Mathematics and Mathematical Sciences 11
For J ′′2 , we obtain
J ′′2 ≤ C∞∑
n=1
nβ−r l(n)∞∑
k=n
kr(−q+1)−1∞∑
j=1
(�Inj)j−rq
(j+1)(k+1)∑
i=2(k+1)
E|X|qI(ir < |X| ≤ (i + 1)r)
≤ C∞∑
n=1
nβ−r l(n)∞∑
k=n
kr(−q+1)−1∞∑
i=2(k+1)
E|X|qI(ir < |X| ≤ (i + 1)r) ∞∑
j=[i(k+1)−1]−1
(�Inj)j−rq
≤ C∞∑
n=1
nβ−r l(n)∞∑
k=n
kr(−q+1)−1∞∑
i=2(k+1)
nr+αir(1−q)k−r(1−q)E|X|qI(ir < |X| ≤ (i + 1)r)
= C∞∑
k=1
k−1∞∑
i=2(k+1)
ir(1−q)E|X|qI(ir < |X| ≤ (i + 1)r) k∑
n=1
nβ+αl(n)
≤ C∞∑
k=1
kβ+αl(k)∞∑
i=2(k+1)
ir(1−q)E|X|qI(ir < |X| ≤ (i + 1)r)
≤ C∞∑
i=4
iβ+α+1+r−rqE|X|qI(ir < |X| ≤ (i + 1)r) ≤ CE|X|1+((β+α+1)/r)l
(|X|1/r
)< ∞.
(3.14)
So∑∞
n=1 nβ−r l(n)
∫∞nr J2dx < ∞. Finally, we prove
∑∞n=1 n
β−r l(n)∫∞nr J3dx < ∞. In fact, noting
1 + (a/r) < s′ < s and β + (qr/2)(1 + (α/r) − s′) < −1, using Markov inequality and (3.1), weget
∞∑
n=1
nβ−r l(n)∫∞
nr
J3dx ≤ C∞∑
n=1
nβ−r l(n)∫∞
nr
( ∞∑
i=1
nrs′x−s′E|aniX|s′)q/2
dx
≤ C∞∑
n=1
nβ−r l(n)nqrs′/2n−r(s′−1)(q/2)nα(q/2)∫∞
nr
x−s′(q/2)dx
≤ C∞∑
n=1
nβ−r+r(q/2)+α(q/2)l(n)nr(−s′(q/2)+1) ≤ C∞∑
n=1
nβ+(qr/2)(1+(α/2)−s′)l(n) < ∞.
(3.15)
Thus, we complete the proof in (a). Next, we prove (b). Note that E|X|1+α/r < ∞ implies that(3.2) holds. Therefore, from the proof in (a), to complete the proof of (b), we only need toprove
I2 = C∞∑
n=1
n−1−r l(n)∫∞
nr
P
{supk≥1
|Snk(x) − ESnk(x)| ≥ n−rxε
2
}dx < ∞. (3.16)
12 International Journal of Mathematics and Mathematical Sciences
In fact, noting β = −1, α + β + 1 > 0, α + β − r < −1 and E|X|1+α/rl(|X|1/r) < ∞. By taking q = 2in the proof of (3.12), (3.13), and (3.14), we get
C∞∑
n=1
n−1+r l(n)∫∞
nr
x−2∞∑
i=1
Ea2niX
2I(|aniX| ≤ n−rx
)dx ≤ C + CE|X|1+(α/r)l
(|X|1/r
)< ∞. (3.17)
Then, by (3.17), we have
I2 ≤ C∞∑
n=1
n−1−r l(n)∫∞
nr
n2rx−2E|Sxn − ESxn|2dx
≤ C∞∑
n=1
n−1+r l(n)∫∞
nr
x−2∞∑
i=1
Ea2niX
2niI(|aniXni| ≤ n−rx
)dx
≤ C∞∑
n=1
n−1+r l(n)∫∞
nr
x−2∞∑
i=1
Ea2niX
2I(|aniX| ≤ n−rx
)dx
+ C∞∑
n=1
n−1−r l(n)∫∞
nr
∞∑
i=1
P{|aniX| > n−rx
}dx
≤ C∞∑
n=1
n−1+r l(n)∫∞
nr
x−2∞∑
i=1
Ea2niX
2I(|aniX| ≤ n−rx
)dx + C < ∞.
(3.18)
The proof of Theorem 2.1 is completed.
Proof of Corollary 2.4. Note that
[∣∣∣∣∣
∞∑
i=1
aniXni
∣∣∣∣∣ − ε
]+≤[supk≥1
∣∣∣∣∣
k∑
i=1
aniXni
∣∣∣∣∣ − ε
]+. (3.19)
Therefore, (2.8) and (2.9) hold by Theorem 2.1.
Proof of Corollary 2.5. By applying Theorem 2.1, taking β = p − 2, ani = n−1/t for 1 ≤ i ≤ n,and ani = 0 for i > n, then we obtain (2.10). Similarly, taking β = −1, ani = n−1/t for 1 ≤ i ≤ n,and ani = 0 for i > n, we obtain (2.11) by Theorem 2.1.
Proof of Corollary 2.6. Let Xni = Yi and ani = n−1/t∑nj=1 ai+j for all n ≥ 1, −∞ < i < ∞.
Since∑∞
−∞ |ai| < ∞, we have supi|ani| = O(n−1/t) and∑∞
i=−∞ |ani| = O(n1−1/t). By applyingCorollary 2.4, taking β = (r/t) − 2, r = 1/t, α = 1 − (1/t), we obtain
∞∑
n=1
n(r/t)−2−(1/t)l(n)E
[∣∣∣∣∣
n∑
i=1
Xi
∣∣∣∣∣ − εn1/t
]+=
∞∑
n=1
nβl(n)E
[∣∣∣∣∣
∞∑
i=−∞aniXni
∣∣∣∣∣ − ε
]+< ∞, ∀ε > 0.
(3.20)
Therefore, (2.12) and (2.13) hold.
International Journal of Mathematics and Mathematical Sciences 13
Acknowledgment
The paper is supported by the National Natural Science Foundation of China (no. 11271020and 11201004), the Key Project of Chinese Ministry of Education (no. 211077), the NaturalScience Foundation of Education Department of Anhui Province (KJ2012ZD01), and theAnhui Provincial Natural Science Foundation (no. 10040606Q30 and 1208085MA11).
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